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A general framework for multi-step ahead adaptive conformal heteroscedastic time series forecasting

Martim Sousa, Ana Maria Tomé, José Moreira

TL;DR

AEnbMIMOCQR presents a model-agnostic framework that delivers distribution-free, multi-step ahead prediction intervals for volatile time series by integrating adaptive conformal inference (ACI), ensemble batch conformalized quantile regression (EnbCQR), and a multi-input multi-output (MIMO) forecasting strategy. The method uses a sliding window of non-conformity scores and horizon-wise CP corrections to achieve near-nominal coverage across $H$ steps while adapting quickly to distribution shifts and heteroscedasticity, without data splitting. Empirical results on real (NN5) and synthetic datasets show improved coverage accuracy and shorter, input-dependent interval widths relative to EnbPI, EnbCQR, and ARIMA, with MIOU indicating better conditional nominal coverage in the synthetic setting. The work extends CP to multivariate, multi-step forecasting settings and provides open-source code, highlighting practical impact in domains with evolving data distributions and uncertain futures, such as finance, energy, and urban planning.

Abstract

This paper introduces a novel model-agnostic algorithm called adaptive ensemble batch multi-input multi-output conformalized quantile regression (AEnbMIMOCQR} that enables forecasters to generate multi-step ahead prediction intervals for a fixed pre-specified miscoverage rate in a distribution-free manner. Our method is grounded on conformal prediction principles, however, it does not require data splitting and provides close to exact coverage even when the data is not exchangeable. Moreover, the resulting prediction intervals, besides being empirically valid along the forecast horizon, do not neglect heteroscedasticity. AEnbMIMOCQR is designed to be robust to distribution shifts, which means that its prediction intervals remain reliable over an unlimited period of time, without entailing retraining or imposing unrealistic strict assumptions on the data-generating process. Through methodically experimentation, we demonstrate that our approach outperforms other competitive methods on both real-world and synthetic datasets. The code used in the experimental part and a tutorial on how to use AEnbMIMOCQR can be found at the following GitHub repository: https://github.com/Quilograma/AEnbMIMOCQR.

A general framework for multi-step ahead adaptive conformal heteroscedastic time series forecasting

TL;DR

AEnbMIMOCQR presents a model-agnostic framework that delivers distribution-free, multi-step ahead prediction intervals for volatile time series by integrating adaptive conformal inference (ACI), ensemble batch conformalized quantile regression (EnbCQR), and a multi-input multi-output (MIMO) forecasting strategy. The method uses a sliding window of non-conformity scores and horizon-wise CP corrections to achieve near-nominal coverage across steps while adapting quickly to distribution shifts and heteroscedasticity, without data splitting. Empirical results on real (NN5) and synthetic datasets show improved coverage accuracy and shorter, input-dependent interval widths relative to EnbPI, EnbCQR, and ARIMA, with MIOU indicating better conditional nominal coverage in the synthetic setting. The work extends CP to multivariate, multi-step forecasting settings and provides open-source code, highlighting practical impact in domains with evolving data distributions and uncertain futures, such as finance, energy, and urban planning.

Abstract

This paper introduces a novel model-agnostic algorithm called adaptive ensemble batch multi-input multi-output conformalized quantile regression (AEnbMIMOCQR} that enables forecasters to generate multi-step ahead prediction intervals for a fixed pre-specified miscoverage rate in a distribution-free manner. Our method is grounded on conformal prediction principles, however, it does not require data splitting and provides close to exact coverage even when the data is not exchangeable. Moreover, the resulting prediction intervals, besides being empirically valid along the forecast horizon, do not neglect heteroscedasticity. AEnbMIMOCQR is designed to be robust to distribution shifts, which means that its prediction intervals remain reliable over an unlimited period of time, without entailing retraining or imposing unrealistic strict assumptions on the data-generating process. Through methodically experimentation, we demonstrate that our approach outperforms other competitive methods on both real-world and synthetic datasets. The code used in the experimental part and a tutorial on how to use AEnbMIMOCQR can be found at the following GitHub repository: https://github.com/Quilograma/AEnbMIMOCQR.
Paper Structure (27 sections, 3 theorems, 25 equations, 7 figures, 4 tables, 4 algorithms)

This paper contains 27 sections, 3 theorems, 25 equations, 7 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Defining quality in prediction intervals
  3. Literature review
  4. Contributions
  5. Paper outline
  6. Background
  7. Forecasting strategies
  8. Recursive
  9. MIMO
  10. Conformal prediction
  11. Conformalized quantile regression
  12. Conformal prediction for time series
  13. Adaptive conformal inference
  14. AEnbMIMOCQR
  15. AEnbMIMOCQR for multivariate time series
  16. ...and 12 more sections

Key Result

Theorem 1

Let $(\bm{X_1},Y_1),...,(\bm{X_{n}},Y_{n})$ be exchangeable random vectors with no ties almost surely drawn from a distribution P, additionally if for a new pair $(\bm{X_{n+1}},Y_{n+1})$, $(\bm{X_1},Y_1),...,(\bm{X_{n+1}},Y_{n+1})$ are still exchangeable, then by constructing $C_{1-\alpha}(\bm{X_{n+

Figures (7)

  • Figure 1: Marginally valid PIs for $\alpha=0.01$.
  • Figure 2: Conditionally valid PIs for $\alpha=0.01$.
  • Figure 3: Marginally valid PIs for $\alpha=0.1$.
  • Figure 4: Conditionally valid PIs for $\alpha=0.1$.
  • Figure 5: Naive conformal PIs for $\alpha=0.01$.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 1: Marginal validity (also known as marginal coverage)
  • Definition 2: Conditional validity (also known as conditional coverage)
  • Theorem 1: Marginal coverage guarantee
  • Definition 3: Exchangeability
  • Lemma 1
  • proof
  • Theorem 2: Marginal coverage guarantee
  • proof